Accurate nonlinear measurements on a device under test performed by a measurement system, for example a Vector Network Analyzer (VNA), require relative and absolute error correction to be performed on the measurement equipment itself.
FIG. 1 is a schematic diagram of a traditional four-wave reflectometer 101 commonly used in measurement systems, for example a VNA. The VNA uses an RF source 109 and a source transfer switch 105 to characterize a two port electrical system, referred to as a device under test (DUT) 103. A local oscillator (LO) 107 is identified in the four-wave reflectometer 101. The VNA is able to measure voltage waves of the four-wave reflectometer in the forward and reverse stimulation directions simultaneously. The reflectometers separate the incident and reflected voltage waves of the DUT 103. These voltage waves are then corrected so that the actual performance (using Scattering Parameters or absolute voltage waves) of the DUT 103 can be accurately measured.
In FIG. 1, the variables used are described as:
axy=the incident voltage wave (independent variable); and
bxy=the reflected voltage wave (dependent variable), where x is the port number and y is the error term designation.
Scattering Parameters (“S-parameters”) are properties used in electrical systems to describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by small signals. The reflectometer 101 can be characterized using S-parameters described in Equations 1.
                                                                                                                                                                            ⁢                                                                  b                        1                        1                                            =                                                                                                    a                            1                            1                                                    ⁢                                                      S                            11                                                                          +                                                                              a                            2                            1                                                    ⁢                                                      S                            12                                                                                                                                                                                                                                      b                      2                      1                                        =                                                                                            a                          2                          1                                                ⁢                                                  S                          22                                                                    +                                                                        a                          1                          1                                                ⁢                                                  S                          21                                                                                                                                                                                                            [                                                                                                    b                        1                        1                                                                                                                                                b                        2                        1                                                                                            ]                            =                                                [                                                                                                              S                          11                                                                                                                      S                          12                                                                                                                                                              S                          21                                                                                                                      S                          22                                                                                                      ]                                ⁡                                  [                                                                                                              a                          1                          1                                                                                                                                                              a                          2                          1                                                                                                      ]                                                                                        Equations        ⁢                                  ⁢        1            
The DUT 103 S-parameters can be calculated from the corrected voltage waves. A corrected voltage wave is one where the measurement system's systematic errors have been removed by performing a calibration and error correction on un-corrected voltage waves, resulting in accurate voltage waves from the DUT. To apply the S-parameters to the DUT, all four of the corrected independent and dependent voltage waves are measured, and then applied to the generalized S-parameter matrix formulations in Equations 1.
Imperfections in the VNA hardware make it necessary to perform error correction to get an accurate representation of the actual voltage waves of the DUT.
A thorough way to characterize the electrical system is through error models. Error models account for systematic errors. Vector error correction is an accurate form of error correction as it accounts for major sources of systematic error.
Vector error correction is a process of characterizing systematic errors by measuring known calibration standards, and then removing the effects of these errors from subsequent measurements. Vector error correction is an example of relative error correction.
A majority of VNAs have a 12-term error model for error correction built into the VNA. The 12-term error model is divided into two sections: a forward and reverse error model. Two-port calibration usually requires twelve measurements on four known hardware standards, short-open-load-through or SOLT. The two-port calibration quantifies twelve systematic error correction terms that are used for subsequent measurements.
An 8-term error model is another generalized error model to correct systematic errors. A benefit of using this model is the ability to supply a signal on both ports at the same time without affecting the error correction process. Any differences to the match seen at the source with respect to the match seen at the terminating load are measured and accounted for. This is performed with all four incident and reflected waves. This identifies the 8-term error model as a single mathematical model that can account for measurements when either port 1, port 2, or both port 1 and 2 are supplying a signal when making both forward and reverse measurements. Error adapters or error correction terms on each side (port 1 and port 2) of the measurement system relate the measured waves to the corrected waves and have no implied relationship to the direction of device stimulation or changes in source or load match of the VNA.
A third error correction model, the 16-term error model, can also be used to correct the systematic errors in a measurement system. However, the 16-term error model is an elaborate process and is difficult to implement practically.
As mentioned above, VNAs utilize vector error correction to correct the systematic measurement errors. In vector error correction, the input frequency to the DUT 103 is the same as the output frequency of the DUT 103. The error correction (typically 12-term) is applied to ratios of measurements (S11, S21, etc).
The 12 and 8-term error models are implicitly known as linear models.
Present error correction is insufficient to meet the needs of characterizing complex electrical systems that exhibit nonlinear behavior. One challenging aspect of measurements with the VNA is to get error corrected nonlinear measurements for the DUT 103.
Error correction for nonlinear measurements can be described as two types: relative and absolute. With relative error correction, linear systematic measurement errors are adjusted for by the vector error correction or by using hardware standards common in the art.
Absolute error correction relies on the absolute amplitude of each measured frequency component. In addition to this, the cross-frequency phase of each frequency component is related to a common time base. The cross-frequency phase is the relative phase difference between each frequency in the measured spectrum. Relating the phase in this way provides a phase difference between all the measured frequency components.
Contemporary VNAs do not accurately measure the nonlinear behavior of the DUT 103 wherein the absolute amplitude and cross-frequency phase of the measured spectral content is required. As a result, a time domain signal representation cannot be constructed accurately from the frequency domain measurements of the waveforms emanating from a nonlinear DUT 103. Additionally, this lack of accuracy inhibits the generation of precise nonlinear behavior models, primarily for use in simulation and design.
Accordingly, a need exists to determine and characterize an electrical system accurately.